To determine which of the given series converges, we need to identify the common ratio [tex]\( r \)[/tex] of each series and check if it satisfies the condition for convergence of a geometric series, which is [tex]\( |r| < 1 \)[/tex].
1. First series: [tex]\( 0.02 + 0.02 + 0.02 + 0.02 + \ldots \)[/tex]
- This is an arithmetic series because each term is the same, not a geometric series.
2. Second series: [tex]\( 4 + 0.08 + 0.0016 + 0.000032 + \ldots \)[/tex]
- To determine the common ratio [tex]\( r \)[/tex], we divide the second term by the first term:
[tex]\[
r = \frac{0.08}{4} = 0.02
\][/tex]
- The common ratio [tex]\( r \)[/tex] is [tex]\( 0.02 \)[/tex].
- Since [tex]\( |0.02| < 1 \)[/tex], this series converges.
3. Third series: [tex]\( 4 + 80 + 1,600 + 32,000 + \ldots \)[/tex]
- To determine the common ratio [tex]\( r \)[/tex], we divide the second term by the first term:
[tex]\[
r = \frac{80}{4} = 20
\][/tex]
- The common ratio [tex]\( r \)[/tex] is [tex]\( 20 \)[/tex].
- Since [tex]\( |20| > 1 \)[/tex], this series does not converge.
4. Fourth series: [tex]\( 0.02 + 0.04 + 0.08 + 0.16 + \ldots \)[/tex]
- To determine the common ratio [tex]\( r \)[/tex], we divide the second term by the first term:
[tex]\[
r = \frac{0.04}{0.02} = 2
\][/tex]
- The common ratio [tex]\( r \)[/tex] is [tex]\( 2 \)[/tex].
- Since [tex]\( |2| > 1 \)[/tex], this series does not converge.
Among the given series, only the second series [tex]\( 4 + 0.08 + 0.0016 + 0.000032 + \ldots \)[/tex] has a common ratio [tex]\( |r| < 1 \)[/tex]. Therefore, it is the only geometric series that converges.
